Perspectivity

Remarks

The map is a bijection from the points of $\ell$ to the points of $m$; its inverse is the perspectivity from $m$ to $\ell$ with the same centre. It fixes the common point of $\ell$ and $m$. A composition of finitely many perspectivities is a projectivity. Dually, let $A$ and $B$ be points and $a$ a line through neither. The perspectivity with axis $a$ sends each line $x$ through $A$ to the line through $B$ and the point $x \cap a$.

Example: in the real affine plane, completed to $\mathbb{P}^2(\R)$, let $\ell$ be $y = 1$, let $m$ be $y = 2$, and let $O = (0, 0)$. The perspectivity sends $(t, 1)$ to $(2t, 2)$. Check: the line through $O$ and $(t, 1)$ is $\set{(st, s)}$, and it meets $y = 2$ at $s = 2$. The common point of $\ell$ and $m$ is their point at infinity, which is fixed.

Non-example: if the centre $O$ lies on $\ell$, the rule fails at $X = O$, since there is no line $OX$ for $X = O$.